A friend broke a string on her bowed psaltery (a small zither-like instrument) and needs to replace it. Problem: what gauge? The source of the instrument is unknown and it didn't come with spare wire or documentation.

Will she have to take the broken string to a music shop and ask for help with identification (maybe compare by visual inspection with other strings they have)? Is there some measurement that someone with ordinary household tools can take to figure it out?

Note that unlike guitar strings, there's nothing particularly special about the strings for a psaltery -- her best bet, once she knows the gauge, will be to just buy a spool of wire. (I doubt anybody even sells "psaltery strings".)

Some electricians' wire gauges may get down to the diameters that you need, but the gauge would have to get down to 30 gauge or smaller; note that 31 or 32 gauge corresponds to an electric guitars smallest strings.

If a micrometer is a "household tool" you can use that to measure the diameter of the wires (some people with home workshops have them). As long as you are using the same type of material (e.g. steel), getting similar diameter wires should get tuned to similar tensions.

However, the key feature that you need to measure is the mass per unit length (not the diameter per se) so you could:

Very precisely measure a length of wire,

Very precisely measure the mass of the wire,

Find other wire with the same mass per unit length.

The actual diameter of the wire is not directly important in affecting the
pitch as a function of tension.* However, it might be hard to get an accurate enough reading from a normal household (kitchen) scale for the lengths of wires that are on a psaltery.

*For example, two different wires with the same
mass/unit-length but with different diameters (this will occur if they are made of different materials) need to be brought to the same tension in order to achieve the same pitch.

A guitar-shop repair person ought to be able to compare the broken string to strings they have in stock to determine the correct diameter.

Once you know the exact diameter, you could probably purchase individual guitar strings of that diameter, quite inexpensively. You could cut a guitar string down to the length needed for the bowed psaltery.

Also, have you looked online for folk music shops that specialize in instruments like the bowed psaltery? I found this link for a company that sells "12-gauge" bowed psaltery string wire in 20-foot and 100-foot spools. Perhaps if you contact such a company, they could tell you if there are different gauges in use and if their gauge would work for your brand of bowed psaltery or not.